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September 10, 2025Groups Geometry and DynamicsOpen Access

Entropy, ultralimits and the Poisson boundary

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The introduction of ultralimits in measure-preserving actions of a group provides insights into invariant measures.
On amenable groups, nearly invariant measures exist with respect to Kullback–Leibler divergence, showing their connection to trivial boundaries.
For free groups, the infimum of the Furstenberg entropy can differ significantly depending on the chosen measure, revealing complex entropy dynamics.
While uniform measures yield consistent values, general measures provide varied Furstenberg entropy outcomes relative to their boundaries.

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The study demonstrates ultralimits of measure-preserving actions in groups, suggesting connections to Furstenberg–Poisson boundaries.

Sayag et al. (2025) studied this question.

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